A set of data is normally distributed with a mean of 100 and a standard deviation of 15. What is the probability that a randomly selected data point will be between 85 and 115?
Answer
Step 4: The probability that the data point is between 85 and 115 is the difference between the probabilities of the two z-scores, which is \(0.8413 - 0.1587 = 0.6826\).
Step 1 ：Step 1: We first need to convert the given data points to z-scores. The formula for finding the z-score is \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 2 ：Step 2: For 85, the z-score is \(z = \frac{85 - 100}{15} = -1\). For 115, the z-score is \(z = \frac{115 - 100}{15} = 1\).
Step 3 ：Step 3: We then use the standard normal distribution table to find the probability for the given z-scores. The probability for z = -1 is 0.1587 and the probability for z = 1 is 0.8413.
Step 4 ：Step 4: The probability that the data point is between 85 and 115 is the difference between the probabilities of the two z-scores, which is \(0.8413 - 0.1587 = 0.6826\).
